Brent oil holds a significant position in the global energy market, as oil prices in many regions are indexed to it. Therefore, forecasting the future price of Brent oil is of great importance. In recent years, artificial intelligence techniques have been widely applied in modeling and prediction tasks. In this study, an Adaptive Neuro-Fuzzy Inference System (ANFIS), a well-established AI approach, was employed for the time-series forecasting of Brent oil prices. To ensure effective learning and improve prediction accuracy, ANFIS was trained using nine different metaheuristic algorithms: Artificial Bee Colony (ABC), Selfish Herd Optimizer (SHO), Biogeography-Based Optimization (BBO), Multi-Verse Optimizer (MVO), Teaching–Learning-Based Optimization (TLBO), Cuckoo Search (CS), Moth Flame Optimization (MFO), Marine Predator Algorithm (MPA), and Flower Pollination Algorithm (FPA). Symmetric training procedures were applied across all algorithms to ensure fair and consistent evaluation. The analyses were conducted on the lowest and highest daily, weekly, and monthly Brent oil prices. Mean squared error (MSE) was used as the primary performance metric. The results showed that all algorithms achieved effective prediction performance. Among them, BBO and TLBO demonstrated superior accuracy and stability, particularly in handling the complexities of Brent oil forecasting. This study contributes to the literature by combining ANFIS and metaheuristics within a symmetric framework of experimentation and evaluation.

In this paper, we introduce the Pell-Lucas and the symmetric Pell-Lucas matrices. The study delves into the linear algebra aspects of these matrices, analyzing their mathematical properties and relationships. We construct decompositions for both the Pell-Lucas matrix and its inverse matrix. We present the Cholesky factorization of the symmetric Pell-Lucas matrices. Furthermore, we derive some valuable identities and bounds for the eigenvalues of these symmetric matrices through the application of majorization notation.

In this work, we explore compositions of positive integers into parts consisting of 1 and 3 s, highlighting their close connection to the Narayana numbers. We also examine palindromic compositions of this type and introduce arithmetic functions that reflect their structural properties. Furthermore, the Woon and PI tree structures related to the Narayana numbers are constructed and visually illustrated. To provide an intuitive understanding, we present tiling models corresponding to these compositions and use them to derive a variety of combinatorial identities. Moreover, we extend our approach to tiling models associated with Narayana polynomials, establishing additional identities through this framework. We also introduce an n ‐color extension of these compositions, exploring how the additional coloring dimension affects their combinatorial properties. Finally, we consider compositions with 1 and 3 s and derive several related identities that further illustrate their relationship with the Narayana numbers.

Risk theory has always played a significant role in mathematical finance and actuarial sciences. A novel approach to the risk theory of non-life insurance is quantum mechanics. To compute finite-time non-ruin probability, I introduce the quantum mechanics formalism in discrete space and continuous space with the appropriately chosen Hamiltonian. By using the quantum mechanics approach and the stochastic method, the non-ruin operator is defined, and tensor products of operator concepts are presented for several examples. In this paper, Dirac notations are operated to find the Hamiltonian matrix with the eigenvector basis for two and three-state cases, and its tensor product version with a change of basis.

Metaheuristic optimization algorithms have become one of the most important research areas for solving complex, nonlinear, and high-dimensional optimization problems. These algorithms are widely used in many application domains such as engineering, artificial intelligence, energy systems, healthcare, communication technologies, manufacturing, data science, and industrial applications. Their strong exploration and exploitation capabilities enable them to provide effective solutions for optimization problems where classical optimization methods are insufficient. In addition, many metaheuristic optimization algorithms attempt to establish a balanced and adaptive search behavior between exploration and exploitation processes, reflecting an important form of search symmetry during the optimization process. This systematic review presents a comprehensive analysis of metaheuristic optimization algorithms based on 515 distinct metaheuristic optimization algorithms. In this study, the algorithms are systematically classified into eleven major categories according to their inspiration sources and dominant search behaviors: Evolutionary-Based, Swarm Intelligence-Based, Bio-Inspired, Ecological/Environmental-Based, Physics-Based, Mathematics-Based, Human-Based, Game/Sport-Based, Chemical and Molecular-Based, Music-, Art-, and Culture-Inspired, and Hybrid and Advanced Metaheuristics. The review analyzes the historical evolution, inspiration diversity, search structures, symmetry-oriented search balance, and development trends of modern metaheuristic optimization algorithms. This study also provides a broader understanding of how metaheuristic optimization algorithms have evolved structurally and conceptually over time. In addition, the study highlights the importance of balanced and adaptive search mechanisms for maintaining optimization efficiency under complex problem conditions. The findings show that the number and diversity of metaheuristic optimization algorithms have increased significantly, especially after 2015. Population-based search structures remain the dominant optimization approach in the literature, while hybrid optimization methods, adaptive search mechanisms, and artificial intelligence integration have become major research trends in recent years. The study also discusses important challenges in the field, including algorithm similarity, benchmark standardization, theoretical limitations, and fair performance evaluation.